Question

34 The midpoint P of the line segment joining the points A(-10, 4) and
B(-2, 0) lies on the line segment joining the points C(-9, 4) and
D(-4, y). Find the ratio in which P divides CD. Also, find the value of y.
[CBSE 2014]​

Answer 1

Answer:-

The midpoint P of a line segment joining the points A( - 10 , 4) and B( - 2 , 0) lies on the line segment joining the C( - 9 , 4) and D( - 4 , y).

We know that,

Co - ordinates of Mid Point of a line segment joining the points (x₁ , y₁) , (x₂ , y₂) are :

$\: \implies \sf \large{ (x \: , \: y) = \bigg( \dfrac{x_1 + x_2}{2} \: \: ,\: \: \dfrac{y_1 + y_2}{2} \bigg)}$

Let,

• x₁ = - 10
• y₁ = 4
• x₂ = - 2
• y₂ = 0

So,

$\: \implies \sf \: P(x \: , \: y) = \bigg( \dfrac{ - 10 - 2}{2} \: \: , \: \: \dfrac{4 + 0}{2} \bigg) \\ \\ \implies \sf \: p(x \:, \: y) = \bigg( \dfrac{ - 12}{2} \: \: , \: \: \dfrac{4 }{2} \bigg) \\ \\ \implies \boxed{ \sf \: p(x \:, \: y) = ( - 6 \: ,\: 2)}$

Now,

P divides CD in a certain ratio , so we have to find the ratio and the value of y.

Using section formula;

The co - ordinates of a point which divides the line segment joining the points (x₁ , y₁) , (x₂ , y₂) in the ratio m : n are :

$\implies \boxed{ \sf \: (x \:, \: y) = \bigg( \dfrac{mx_2 + nx_1}{m + n} \: \: , \: \: \dfrac{my_2 + ny_1}{m + n} \bigg)}$

Let,

• x = - 6

• y = 2

• x₁ = - 9

• x₂ = - 4

• y₁ = 4

• y₂ = y

• m = m

• n = n

So,

$\: \implies \sf \: ( - 6 \: , \: 2) = \bigg( \dfrac{(m)( - 4) + (n)( - 9) }{m + n} \: \: ,\: \: \dfrac{(m)(y)+ (n)(4)}{m + n} \bigg) \\ \\ \implies \sf \: ( - 6 \: , \: 2) = \bigg( \dfrac{ - 4m - 9n}{m + n} \: \: , \: \: \dfrac{ym + 4n}{m + n} \bigg) \\ \\ \implies \sf \: - 6 = \dfrac{ - 4m - 9n}{m + n} \\ \\ \implies \sf \: - 6(m + n) = - 4m - 9n \\ \\ \implies \sf \: - 6m - 6n + 4m = - 9n \\ \\ \implies \sf - 2m = - 9n + 6n \\ \\ \implies \sf - 2m = - 3n \\ \\ \implies \sf \frac{ - 2 \times m}{ - 3 \times n} = 1 \\ \\ \implies \sf \frac{m}{n} = \frac{3}{2} \\ \\ \implies \boxed{ \sf \: m : n = 3 : 2}$

Similarly,

$\implies \sf \: 2 = \dfrac{ym + 4n}{m + n}$

Substitute the values of m , n here.

$\implies \sf \: 2(3 + 2) = y(3) + 4(2) \\ \\\implies \sf 10 - 8 = 3y \\ \\ \implies \boxed{ \sf \: \frac{2}{3} = y}$

∴

• P divides CD in the ratio 3 : 2.
• y = 2/3.